Confidenc Interval(16 3) Biostatistics

8/8/2019 Confidenc Interval(16 3) Biostatistics

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Estimation of Population Means:Point Estimation and Confidence

Interval


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Statistics

Descriptive Inferential

Estimation Hypothesis testing

Point estimateInterval estimates

(ConfidenceInterval)


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Types of Estimators

Point Estimator

- It gives a single value as an estimate of the

parameter of interest

Interval Estimator

- It specifies a range of values of the parameter and our

confidence that the parameter value is in that range


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Point Estimation

A point estimate of the population parameter is the samplestatistic computed from a random sample drawn from the

population under study.

Certain sample statistic are good point estimators for certain

parameters-G ----- Estimates ----- ----- Estimates ----- W

Sample mean is a statistic that varies from sample to sample If the investigator had repeated the experiment, he would have

found a range of sample means, any one of which would be a

point estimate of the population mean.


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APoint Estimate is a single number, How much uncertainty is associated with a point estimate of a population

parameter?

The point estimate method fails to indicate how close the estimate isto population parameter. This flaw can be remedied by use of aconfidence interval estimate (CI).

An interval estimate provides more information about apopulation characteristic than does a point estimate. It provides

a confidence level for the estimate. Such interval estimates arecalled Confidence Intervals

Point Estimate

Lower

Confidence

Limit

Width ofconfidence interval

Upper

Confidence

Limit


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Interval Estimation

It is the interval of numbers in which we have a specified degree of

assurance that the value of the parameter can be found.

The level of confidence tells the probability the method produced aninterval that includes the unknown parameter

Gives information about closeness to unknown population

parameters

Stated in terms of level of confidence. (Can never be 100%

confident)


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Confidence interval for population

parameter A confidence interval is a formula that tell us

how to use sample data to calculate an interval

that estimate a population parameter e.g.

population mean ().

The confidence level is the confidence

coefficient expressed as a percentage i.e.

(1- )%


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Empirical Rule Definition

For data sets having a normal bell-shapeddistribution, the following properties apply:

About 68% of all values fall within 1 standard

deviation of the mean

About 95% of all values fall within 2 standard

deviation of the mean

About 99.7% of all values fall within 3 standarddeviation of the mean.


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Confidence Interval

The general formula for all confidence intervals is equal to:

Point Estimate (Critical Value) * (Standard Error)

Now using the Empirical Rule for the normal

distribution we know that the interval X + 2 /n , or

more precisely, the interval X + 1.96 /n includes 95%of Xs in the repeated sampling.


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Consider a 95% confidence interval:

Z= -1.96 Z= 1.96

.05!E

.0252

! .025

2

!

Point EstimateLowerConfidenceLimit

UpperConfidenceLimit

Point Estimate

0

.951 !E .0252/ !E

.475.475

Z

l u


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Confidence Intervals

Formula:

Steps:

1.Calculate the sample statistic to use as an estimate of

the population parameter

2.Calculate the lower (LL) and the upper limits (UL) of

the confidence interval

n

ZXW

Ey

2/

eQe

nZX

/

Wy

E 2


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Determination of In order to construct an interval estimate, it is necessary

to obtain some estimate of , the variability of thepopulation from which the sample is drawn.

This is required to obtain an estimate of the standard

error of the sample mean

Generally, the sample standard deviation s is used as anestimate of .

For a small sample, where n < 30, the t-distributionshould be used, again using s as an estimate of .

n

x

WW !


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Level of Confidence

Probability that the unknown population parameter is in the

confidence interval in 100 trials. Denoted (1 - ) % = level

of confidence e.g. 90%, 95%, 99%

Is Probability that the parameter is not within the interval

in 100 trials


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Selecting a confidence level

There is no one confidence level that isappropriate for all circumstances.

Greater confidence level means greater certaintythat the interval estimate of actually contains

. But for 99% or 99.9% confidence level, theinterval may be very wide.

Smaller confidence levels (eg. 80% or 90%)

produce smaller margins of error and seeminglymore precise interval estimates, but they are lesslikely to contain .

By tradition, the default level is 95%.


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Interpretation

The interpretation of the confidence interval is very

important. Basically it means that upon taking a sample of

size n repeatedly and constructing the interval

X + 1.96 /n each time, we would expect the populationmean Q to fall within the interval 95% of the time .


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Interpretation of a Confidence

Interval for Population Mean (

) We can be 100(1-)confident that lies between the lower andupper bounds of the confidence interval.

In other way, it means that upon taking a sample of size

n repeatedly and constructing the interval X + 1.96/n each time, we would expect the population meanQ to fall within the interval 95% of the time

The values are called lower and upper 100(1-)%

confidence limits.


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Commonly used values of Z/2

Confidence level

100 (1-) 2

Z2

90% 0.10 0.05 1.645

95% 0.05 0.025 1.96

99% 0.01 0.005 2.575


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Example 2

If we wish to estimate the mean VO2 uptake for a

population of joggers based on a random sample of 100

joggers, we could use the 95% confidence interval for Q.From our random sample of 100 joggers we know that X =

47.5 ml/kg and S = 4.8 ml/kg. A 95% Confidence Interval

(C.I.) of Q isX + 1.96 S /n or 47.5 + 1.96 ( 4.8)/10

47.5 + 0.94 or ( 46.56, 48.44)

The values 46.56 and 48.44 are the lower and upper 95%

confidence limits. Interpretation: We are 95% confident that in the long run

the intervals constructed in such a way will contain thepopulation mean Q.


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Example 3

If we wish to estimate the mean VO2 uptake for a

population of joggers based on a random sample of 100

joggers, we could use the 99% confidence interval for Q.From our random sample of 100 joggers we know that X =

47.5 ml/kg and S = 4.8 ml/kg. A 99% Confidence Interval

(C.I.) of Q isX + 2.575 S /n or 47.5 + 2.575 ( 4.8)/10

47.5 + 1.24 or ( 46.26, 48.74)

The values 46.26 and 48.74 are the lower and upper 99%

confidence limits. Interpretation: We are 99% confident that in the long run

the intervals constructed in such a way will contain thepopulation mean Q.


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Width of a Confidence Interval

The width of any confidence interval is the difference

between the upper confidence limit and the lower

confidence limit .

The width of a confidence interval represent theaccuracy of estimation .


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Factors Affecting Interval Width

Data Variation

measured by

Sample Size n

Level of Confidence

(1 - )

Confidence Interval Estimate

nZX

/

Wy E 2 eQe

nZX

/

Wy E 2

Narrow widths and high confidence levels are desirable, but

Narrow widths and high confidence levels are desirable, butthese two things affect each otherthese two things affect each other


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Why Narrow Confidence Interval areImportant ?

Narrow confidence intervals are of the greatest value inmaking estimates ,because they allow us to estimate anunknown parameter with little room for error .

Aconfidence interval can be narrowed by:

Increasing the sample size .

Reducing the confidence level (1-)100%

Reducing the source of variability in the observations ,thus

producing less variance .


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Cautions about interval

estimates There are many assumptions involved in intervalestimation: The sample is randomly selected from a population.

The sample size is sufficiently large

The population standard deviation is known or s is a goodestimate of .

The selection of a confidence level is an arbitrary process.

The population is not too skewed

As a result, interval estimates are not precise, but are estimates orapproximations.

Larger n, repeated sampling, comparisons with otherstudies, and careful sampling and survey design andpractice can improve the quality of the estimates.


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95%Confidence Interval

A 95% is the mostfrequent reportedconfidence intervalreported. Not that when

you see certain intervalestimates reported on TV(for example somebusiness or medicalstatistics), the confidence

level is not mentioned butit is under stood that it isbased on a 95%confidence level.

68% CI More Error

Narrow CI

95% CI Medium Error

Narrow CI

99% CI Less Error

Wider CI

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Confidenc Interval(16 3) Biostatistics